The bootstrap is an increasingly popular method for performing statistical inference. This paper provides the theoretical foundation for using the bootstrap as a valid tool of inference for quasi-maximum likelihood estimators (QMLE). We provide a unified framework for analyzing bootstrapped extremum estimators of nonlinear dynamic models for heterogeneous dependent stochastic processes. We apply our results to two block bootstrap methods, the moving blocks bootstrap of Künsch (1989) and Liu and Singh (1992) and the stationary bootstrap of Politis and Romano (1994), and prove the first order asymptotic validity of the bootstrap approximation to the true distribution of QML estimators. Further, these block bootstrap methods are shown to provide heteroskedastic and autocorrelation consistent standard errors for the QMLE, thus extending the already large literature on robust inference and covariance matrix estimation. We also consider bootstrap testing. In particular, we prove the first order asymptotic validity of the bootstrap distribution of a suitable bootstrap analog of a Wald test statistic for testing hypotheses.
